What is the homotopy type of the group of diffeomorphisms of the 4-sphere?
) is known to be a -fold loop space. In particular there is a homotopy-equivalence known as the Cerf-Morlet Comparison theorem where is the group of PL-automorphisms of and is the group of linear automorphisms of . Otherwise there is not much in the literature about . Since it is a group of diffeomorphisms it has the homotopy type of a countable CW-complex. It is unknown whether or not it is connected, or if it has any other non-trivial homotopy or homology groups.
is known to have the homotopy-type of provided by work of Hatcher and Smale respectively. For many of the groups were computed by Kervaire and Milnor, who further related these groups to the homotopy groups of spheres. For the rational homotopy groups of have been computed by Farrell and Hsiang in range . They show .
[B] Budney, R. Little cubes and long knots. Topology. 46 (2007) 1--27.
[FH] Farrell, F.T. Hsiang, W.C. On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds. Proc. Symp. Pure. Math. 32 (1977) 403--415.
[H] Hatcher, A proof of a Smale conjecture, . Ann. of Math. (2) 117 (1983), no. 3, 553--607.
[KS] Kirby, R. Siebenmann, L. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton University Press.
*[S] Smale, S. Diffeomorphisms of the 2-sphere, Proc. Amer. Math. Soc. 10 (1959) 621--626.
* indicates original appearance(s) of problem.